SIGNS AND MEANINGS IN STUDENTS’ EMERGENT ALGEBRAIC ... · The very semiotic nature of thinking and knowledge is addressed in Section 1, where we discuss some problems about signs

LUIS RADFORD

SIGNS AND MEANINGS IN STUDENTS’ EMERGENT ALGEBRAICTHINKING: A SEMIOTIC ANALYSIS ?

ABSTRACT. The purpose of this article, which is part of a longitudinal classroom researchabout students’ algebraic symbolizations, is twofold: (1) to investigate the way students usesigns and endow them with meaning in their very first encounter with the algebraic gen-eralization of patterns and (2) to provide accounts about the students’ emergent algebraicthinking. The research draws from Vygotsky’s historical-cultural school of psychology,on the one hand, and from Bakhtin and Voloshinov’s theory of discourse on the other,and is grounded in a semiotic-cultural theoretical framework in which algebraic thinkingis considered as a sign-mediated cognitivepraxis. Within this theoretical framework, thestudents’ algebraic activity is investigated in the interaction of the individual’s subjectivityand the social means of semiotic objectification. An ethnographic qualitative methodology,supported by historic, epistemological research, ensured the design and interpretation of aset of teaching activities. The paper focuses on the discussion held by a small group ofstudents of which an interpretative, situated discourse analysis is provided. The resultsshed some light on the students’ production of (oral and written) signs and their meaningsas they engage in the construction of expressions of mathematical generality and on thesocial nature of their emergent algebraic thinking.

KEY WORDS: semiotic-cultural approach to algebraic thinking, generalization, signs andmeanings, symbolization, social means of semiotic objectification

INTRODUCTION

Since Davis (1975), Clement (1982), Clement et al. (1981) and other pi-oneer studies on the learning of algebra, a significant number of didacticworks has focused on the investigation of students’ algebraic thinking andthe search of pedagogical means to enhance it. Some of these works havedealt with generalization – a topic that several curricula around the worldencourage as a route to algebra (see e.g. MacGregor and Stacey, 1992).The research results have stressed the fact that generalization of numericalpatterns and the symbolic formulation of relations between variables raisespecific problems for novice students (see e.g. Arzarello, 1991; Arzarelloet al., 1993, 1994a, 1994b; MacGregor and Stacey, 1993; Mason, 1996;? This paper is part of a research program funded by the Social Sciences and Humanities

Research Council of Canada, grant # 410-98-1287.

Educational Studies in Mathematics42: 237–268, 2000.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

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Lee, 1996; Rico et al., 1996). Although particular difficulties experiencedby students have been reported by the aforementioned works and manyothers, it has been recently argued that more research is still needed (Sas-man et al., 1999, p. 161). To go further, we want to add, we need to deepenour own understanding of the nature of algebraic thinking and the way itrelates to generalization.

In this paper, we want to contribute to such an enterprise by interweav-ing classroom observations and theoretical reflections. To do this, we shalladopt a social and semiotic perspective having a central focus of attentionon the students’ understanding and constitution of the meaning of signs asused in algebra (see e.g. Nemirovsky, 1994; Meira, 1996; Vile and Lerman,1996). In more specific terms, the purpose of this paper is:

(1) to investigate the way in which students use signs and endow themwith meaning during their very first tasks related to the algebraicgeneralization of patterns1 and

(2) to provide accounts about the students’ emergent algebraic thinking.

Our approach is underlain by an anthropological view about the natureof thinking. As such, it relates thinking to the social practices from whichit arises. There are, of course, several forms in which to theorize such arelation (see e.g. Chaiklin and Lave, 1993). Ours elaborates this relationin a way that thinking is conceived as a form of social sign-mediated cog-nitive praxis. It is against this anthropological background that, in whatfollows, we shall investigate the two aforementioned objectives. We willscrutinize the use of signs and their meanings in the students’ emergentalgebraic thinking in the intersecting territory of individual subjectivityand the social means of semiotic objectification. In doing so, we departfrom epistemological evolutionary postures and their correlated view ofthinking according to which thinking appears as anacultural, natural, tele-ologically necessary process of abstraction in the intellectual developmentof the students.

The very semiotic nature of thinking and knowledge is addressed inSection 1, where we discuss some problems about signs and their mean-ings in light of our semiotic-cultural theoretical framework. Section 2 dealswith the description of our classroom research program and the differentphases underpinning its ethnographic methodology. Sections 3 to 5 aredevoted to the analysis of a classroom activity about generalization wherewe investigate the novice students’ expression of the general term of ageometrical-numeric pattern, focusing on the discussion held by a smallgroup of students of which we provide an interpretative, situated discourseanalysis. In Section 6, the results of the previous sections are re-examinedat a more general level in order to discuss the emergent students’ algeb-

SIGNS AND MEANINGS IN STUDENTS’ EMERGENT ALGEBRAIC THINKING 239

raic thinking through the prism of our anthropological approach and itssemiotic-cultural theoretical framework.

1. SIGNS AND THEIR MEANING2

Teaching algebra seems to have been a pedagogical problem since An-tiquity. In the introductory note to his monumentalArithmetica, writtenca. 250 AC, Diophantus of Alexandria mentions the discouragement thatthe students usually feel when learning what we now term ‘algebraic tech-niques’ to solve word-problems. Contemporary mathematics curricula tryto offer students some help to develop the algebraic ideas and to acquireand make sense of signs. For example, in the new Ontario Curriculumof Mathematics (Ministry of Education and Training 1997), students areintroduced to a kind of ‘transitional’ language prior to the standard alpha-numeric-based algebraic language and are asked to find the value of ‘∗’ inequations like:∗ + ∗ + 2 = 8 or the value of ‘�’ in equations like 32+� + � = 54. This ‘transitional language’ approach, as any pedagogicalapproach for teaching algebra, relies on specific conceptions about whatsigns represent and the way in which the meaning of signs is elaborated bythe students.

What does a sign represent? This question – also discussed by Mason(1987) – is at the center of the semiotic problem concerning the relationbetween the sign and its signified. Some psychological approaches, follow-ing the distinction of expression vs. content or surface structure vs. deepstructure made in structural linguistics in the 1970s, have elaborated thisproblem in terms of the relation between internal (or mental) represent-ations and external representations (seen as e.g. manipulation of signs orsymbols). The link between both kinds of representations has often beendescribed in terms of the property of a postulated mapping process accord-ing to which the external manipulations somehowreflectthe structures ofmental functioning. Long before the rise of cognitive psychology, Fregeclearly expressed this idea. He said that the structure of propositions ap-pears as a kind of mirror of the structure of thinking (Frege, 1971, p. 214).This position, however, is problematic in that it leaves unanswered thequestion of the nature of the mapping between the sign and the signified.Furthermore, this position is reductive in that, as Meira (1995) pointed out,signs, symbols and mathematical notations are seen as mereaccessoriesfor the construction of concepts.

Theoretical approaches based on the dichotomy internal/external rep-resentations are also problematic in the views that they offer concerningthe objects(e.g. a physical entity) to which the external representations

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refer. The object (or what is sometimes called thereferentand that Stoicphilosophers calledtynchanon) is reified and abstracted from its historicaland cultural context and is conceived as standing in front of the subject asif the object was a mere item in the ecology of the subject.

The last remark has important consequences in the way meaning con-struction is conceptualized. Indeed, as long as the relation subject/objectis seen as a non-culturally mediated, direct one, meaning construction ap-pears to be the result of the relation that the isolated subject entertains withtheahistorical object (for a detailed discussion see Radford, 2000).

In the particular example of the ‘transitional language’ approach men-tioned previously – as well as in other variants of structural approachesto meaning making in elementary algebra with epistemological and theor-etical commitments to the information processing psychology (Kirshner,2001) – meaning is expected to be extracted by the student (at least tosome important extent) from the syntactic structure of the algebraic lan-guage and endless tasks of symbolic manipulation. The understanding ofthe syntactic structure of algebraic language and the meaning of signs is,however, a long process in the students’ ontogenetic trajectory. Students, atthe very beginning, tend to have recourse to other experiential aspects moreaccessible to them than the structural one. In their study about students’understanding of algebraic notations, MacGregor and Stacey (1997, p. 5)notice that some novice students associate letters with numbers accordingto the position in the alphabet (e.g. ‘h’ with 8) and suggest that this is dueto the students’ experience in puzzles and translation into codes activities.We take examples like this as a clear indication that novice students bringmeanings from other domains (not necessarily mathematical domains) intothe realm of algebra. Hence, it seems to us, one of the didactic questionswith which to deal is not really that of the elaboration of catalogues ofstudents’ errors in algebraic manipulations, which may be interesting initself, but that of the understanding how those non-algebraic meaningsare progressively transformed by the students up to the point to attainthe standards of the complex algebraic meanings of contemporary schoolmathematics3.

The theoretical position that we are taking in our research programconcerning, on the one hand, the conceptual aspect of signs (i.e., the sign-signified relationship) and, on the other hand, their signifying feature (i.e.,the one related to their meaning) is based on a general semiotic-culturalconception of cognition having its roots in two basic ideas. The first oneis the Vygotskian idea according to which our cognitive functioning isintimately linked, andaffectedby, the use of signs4. We take signs here notas mere accessories of the mind but as concrete components of ‘menta-


tion’. Following Vygotsky (1962), Vygotsky and Luria (1994), Zinchenko(1985), Geertz (1973), Bateson (1973) and Wertsch (1991), instead ofseeing signs as the reflecting mirrors of internal cognitive processes, weconsider them as tools or prostheses of the mind to accomplish actions asrequired by the contextual activities in which the individuals engage5. Asa result, there is a theoretical shift from what signsrepresentto what theyenableus to do6.

The second basic idea on which our framework is based deals with themeaning of signs and stresses the fact that the signswith which the indi-vidual acts andin which the individual thinks belong to cultural symbolicsystems which transcend the individualqua individual. Signs hence have adouble life. On the one hand, they function as tools allowing the individu-als to engage in cognitive praxis. On the other hand, they are part of thosesystems transcending the individual and through which a social reality isobjectified. The sign-tools with which the individual thinks appear then asframed by social meanings and rules of use and provide the individual withsocial means of semiotic objectification.

In this line of thought, the conceptual and the signifying aspects ofsigns need to be studied in the activity that the signs mediate in accord-ance to specific semiotic configurations resulting from, and interwovenwith, social meaning-making practices and cultural forms of signification(Radford, 1998a).

As a result of our theoretical requirements and our didactic purposes,the classic semiotic triangle, having as its vertexes the sign, its object andits signified, cannot suitably account for the conceptual relations of signsand the aspects related to their meaning. Such semiotic triangles oftenisolate the subject, the object and the act of symbolizing from the otherindividuals and their contextual activities7.

Within the context of the previous discussion, in our classroom re-search program we consider the learning of algebra as the appropriationof a new and specific mathematical way of acting and thinking which isdialectically interwoven with a novel use and production of signs whosemeanings are acquired by the students as a result of their social immersioninto mathematical activities.

Naturally, by this we do not mean that students’ knowledge appropri-ation is achieved through a kind of crude transfer of information com-ing from the teacher8. As we see it, knowledge appropriation is achievedthrough the tension between the students’ subjectivity and the social meansof semiotic objectification. It is in this sense that we will proceed in thenext sections to examine the activity of a small group of students deal-ing with a task about generalization of patterns. In this didactic task, the

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different psychological, epistemological, semiotic and historical perspect-ives, which serve as the basis of our framework (elaborated in detail inRadford, 1998b), play a different role. While we retain the idea of signs asthe concrete components of mentation, as suggested by cultural and Vy-gotskian studies, we borrow from Bakhtin (1986) and Voloshinov (1973)important elements of their theory of dialogue to understand the role ofstudents’ discursive actions and interactions. In this perspective, speech,in the various formats that discourse lodges, is not seen as an exchangeof information. Rather, we ascribe to speech an epistemological role inthe individual’s reconstruction and novel construction of knowledge. (Forother – similar or different – views concerning the epistemological role oflanguage and its relevance for mathematics education see, e.g. Boero etal., 1997, 1998; Bartolini Bussi, 1995, 1998; Bauersfeld, 1995; Cobb etal., 1997; and Steinbring, 1998)9.

2. ABOUT THE METHODOLOGY

The research reported in this paper is part of a three-year longitudinalclassroom research dealing with the students’ learning of algebra. Ourresearch is based on an ethnographic qualitative design (Goetz and Le-Compte, 1984) having the following three-phase architecture: (i) a firstphase, that may be termedingénierie didactique(Artigue, 1988), in whichthe design of the teaching settings is elaborated, (ii) a second phase, inwhich the implementation of those teaching settings in the classroom oc-cur, and, (iii) a third phase, where the research analyses are carried out.The three phases take into account the specificity of the curriculum ofmathematics which functions as a reference point for learning assessmentand further development.

The first phase includes the elaboration of the classroom activities withthe teachers. The activities have been elaborated in such a way that thestudents (who are presently in Grade 8) work together in small groups.Then, the teacher conducts a general discussion allowing the students toexpose, confront and discuss their mathematical methods and solutions.The second phase consists of the implementation and video-taping of theactivities on which the teaching settings are based10. Drawing some meth-odological elements from Discourse Analysis (mainly from the works ofFairclough, 1995; Moerman, 1988 and Coulthard, 1977), the third phaseaddresses the discussion of the video-tapes, their transcription and inter-pretation. Finally, the discussions between the teachers, researcher andresearch assistants provide feedback for the instructional design of theup-coming algebraic curricular units.


Figure 1. Problem discussed by the teacher and the students.

The classroom observations are of two different types: a) the video-taping of two to three co-operative groups of two to four students (de-pending on the mathematical activity) and b) quizzes administered to thewhole class at the end of the teaching units11. Four classes are participat-ing in the longitudinal research program. The four classes come from twodifferent schools (which will be identified as school A and B) situated innorthern Ontario and may be considered as typical schools of medium sizecities in Canadian standards. Although some efforts have been made bythe provincial government over the last few years to improve the qualityof teaching and learning in mathematics (particularly the recent revampingof the curricular teaching content as a result of Ontario’s poor scores innational and international mathematics tests), it is still too early to knowwhat impact this will have.

The schools A and B have an independent curricular schedule, which al-lows us to continually generate feedback from one school to the other. Forinstance, after realizing the enormous difficulties that students in school Ahad in understanding what was expected from them in the algebraic task ofwriting the n-th term of a pattern (Radford, 1999), in school B we decidedto open the classroom activity about generalization with a typical examplethat the teacher discussed with the students before they went to work insmall groups. An overview of the introductory example and an analysis ofthe small-group activity are provided in the next section.

3. THE CLASSROOM ACTIVITY

The typical example that the teacher discussed with the students, prior tothe small groups classroom activity, consisted of a sequence of rectangles(see Figure 1). After counting the number of squares in the first rectangles,the discussion revolved around finding a way to calculate the number of

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squares in rectangles such as the 25th. The students associated the numberof squares to the area of a rectangle and, with the help of the teacher,identified the area with the numerical sequence 1× 3, 2× 4, 3× 5, 4× 6.

The teacher then asked for a formula that would work for any rectangleof the sequence. Through an interactive question-answer process in whichmany students participated, the link between the number of the rectangle inthe sequence and the number of squares on its length, on the one hand, andthe number of squares on its width, on the other hand, became apparent.This led them to the symbolic expressionn × (n + 2), which was thenapplied to a few concrete cases.

Following the classroom discussion of the introductory example, thestudents engaged in a small group classroom activity. In order to explorethe discursively- and symbolically-based students’ semiotic means of ob-jectification in generalization tasks, the activity was divided into threesteps:

(i) an arithmetic investigation,(ii) the expression of generalization in natural language (in the form of a

message), and(iii) the use of standard algebraic symbolism to express generality.

On the first day, the students were presented two problems. We willfocus on the first one (see Figure 2).

In what follows, we shall concentrate on one of our small groups ofstudents. This group (like the others) was made up of 3 students – Anik,Josh and Judith – (naturally not their real names for deontological reasons).

A general overview of the role of transcript analysis and its place in ourlongitudinal classroom research was given in Section 2. For the purposeof the specific aspect of the research that we are reporting here, we needto add that the interpretative transcript analysis, inspired by Coulthard’work (Coulthard, 1977), was carried out in three steps. In the first step, allutterances were treated equally not paying attention to context, intention,and so on. In the second step, the rough material resulting from the firststep was put into categories (see below) then refined into salient segments(Côté et al., 1993) and then contextualized by adding a social interactionistdimension (captured through interpretative comments that we insert in ital-ics in the dialogue, emphasizing communicative aspects in relation to theresearch problem at hand). In the third step, the cadence of the dialoguewas inserted by indicating pauses and verbal hesitations. The whole ofthe three aforementioned steps leads to what we want to term asituateddiscourse analysiswhose elementary unit (i.e. the unit of analysis) wasconstituted by the refined (i.e. contextualized and cadenced) identified sa-lient segments which were managed with theNon-numerical Unstructured


Figure 2. Problem presented to the students during the small group activity.

Data Indexing Searching and Theorizing(Nud·ist) program for qualitativeresearch (Gahan and Hannibal, 1998).

In accordance to our theoretical framework and research purpose, thediscourse analysis seeks to provide explanations about how students cometo use signs and appropriate their meanings in the course of their initiationinto the social practice of algebra. Although it is not possible to list theingredients of meaning in advance, for, as Moerman (1988, p. 7) notices,‘[t]he interpretations that conversants make [of the other conversants’ ut-terances] are, in fact,post facto’, something which makes the territory ofthe encounter between subjectivity and the social means of objectificationvery flexible, we will scrutinize the students’ dialogue in light of theirhandling of the relationship between the particular and the general, itssemiotization in natural language and in standard algebraic signs.

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4. EXPRESSING GENERALITY IN NATURAL LANGUAGE

A quick indication of the difficulty that students experienced in dealingwith the construction of generality in natural language is provided by thelength of the dialogue (measured as the number of lines). While questionsa) and b) were solved in 3 or 4 lines, question c) generally took about 140lines. In the following excerpt, we see how question b) was dealt with:

Line Dialogue/comments

54 Judith: OK(Reads)‘How many circles would figure number 100 have in total?’That would be 100 . . . (reflecting)

55 Anik: . . . (interrupting) 199 because you have 100 on the bottom and 99 on top.56 Josh: (inaudible)57 Judith: Yes(Writes the answer). . . OK.

Given the length limitation of this article, it is impossible to offer a tran-scriptionin extensoof the students’ dialogue concerning question c). Thus,in the next part (Subsection 4.1) we will present and discuss only some ofthe salient segments12, focusing on the way the students linked the generaland the particular.

4.1. The general and the particular: the role of the metaphor

Here is an excerpt of the students’ dialogue:

63 (Josh silently reads the problem. Anik and Judith are looking at him. Then he says. . .) I don’t understand.

64 Anik: (Talking to Josh) OK. You just have to explain to someone, let’s say me(pointing to herself), I don’t know how to do it, like I don’t know. . . like what thisis here (pointing to a figure on the page). You have to show me how I would knowhow many chips to put, like if it said: in figure . . . uh . . . Idon’t know. . . 120, OK?You would have to explain really well why, I mean, how I would. . . /[. . . ]

71 Josh: How would I say that?72 Anik: OK. Alright, look. You say how one has to add (pointing to a figure on the

paper) . . . you always add 1 to the bottom, right? Then you always add 1 to the top(referring to the rule to go from one figure to the next). (When she utters the word‘top’ her hand moves to the top of the figure on the paper.) So it’s always 1,2 . . . waita minute. Do you know what I’m talking about?

73 Josh: Yes.74 Anik: Do you know what it is that I want to say?75 Josh: So, you start by adding . . . how I . . . exactly in a sentence?76 Anik: You could say, uh, the figure . . . OK, say: Let’s say that in figure . . . um

. . . Figure 12 (moving her hands on the desk near the 6 first figures made up ofbingo chips, as if she were touching the hypothetical Figure 12). You’d put 12 chips. . .


77 Josh: on the bottom . . .78 Judith: on the bottom . . .79 Anik: That would go . . .80 Josh: Not on the bottom.81 Judith: Horizontal? /[. . . ]84 Josh: OK. Miss (raising his hand) . . . how would I explain it? I know the answer, it’s

just that . . . /[. . . ]87 Anik: (after being reassured by the teacher about the correct use of the word ‘hori-

zontal’) OK. So you’d put the number of chips horizontally with the figure.

In line 64, Anik rephrases with her own words the instructions about themessage to be written. At the end of line 64, she hypothetically takes therole of the addressee (‘You would have to explain really well why, I mean,how I would . . . ’). Interestingly, in this move, consisting in taking the placeof others and which is essential in social understanding (Astington, 1995),she omits the linguistic expression conveying the generality, i.e., that themessage must say what to do to know how many circles are inany figure.Instead, she takes a concrete figure – Figure 120 – as an example to talkabout the general. In line 71, Josh overtly asks how tosay it. We now seethe enormous difference in talking about obtaining an answer for a specificfigure and saying it for any figure in general (as the student acknowledgesin line 84, when the teacher comes to see the group’s work).

To talk in general terms, they hence take a specific figure, which is Fig-ure 12 from line 76 onwards. Notice, however, that Figure 12 (as well as theaforementioned Figure 120) is not among those made with colored plasticbingo chips that the studentsmaterially have in front of them. Thanks toits ‘unmateriality’, Figure 12 fits the purpose of their reasoning about thegeneral very well.

Nevertheless, Anik and her group-mates are not really talking about theparticular Figure 12, something emphasized by the hypothetical expres-sion ‘Let’s say’ (line 76). This is why they are not strictly counting thenumber of circles in Figure 12. We may say hence that Figure 12 is nottaken literally butmetaphoricallyby the students. In discursively takingan absent albeit specific figure, they talk metaphorically about the generalthroughthe particular13.

4.2. Talking about the metaphor: Deictic and generative functions oflanguage

Of course, Figure 12 is not enough to deal with mathematical generality.The students will now display a range of subtle discursive resources fortheir objectifying process of generality through the metaphorical figure. Aclose look at the dialogue shows that structural elements in reasoning aboutand expressing generality are based upon two key categories of words

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Figure 3. Generative action function and deictic function.

having two different semiotic functions: agenerative action functionandadeictic function(see Figure 3).

Deictic terms are linguistic units (in our example, ‘top’, ‘bottom’) re-ferring to objects in the universe of discourse (in this example, the figuresof the pattern) by virtue of the situation where dialogue is carried out. Itis the contextual circumstances which determine their referents. As such,deictic terms depend heavily on the context (see Nyckees, 1998, p. 242 ff.)and have a particular function in dialogical processes.

The linguistic terms (‘top’, ‘bottom’) through which the deictic func-tion is carried out (lines 72, 77, 78, 80, etc.) – terms which are based onthe geometric shape of the concrete figures – allow them to refer to specificparts of the general mathematical object which, from the students’ point ofview, is being socially and discursively constructed at this precise moment.The interrogative interjection ‘right?’ (line 72) – strategically placed atthis point of the unfolding dialogue and the social process of semioticobjectification – serves as a discursive request for gaining approval.

We term ‘generative action function’ the linguistic mechanisms ex-pressing an action whose particularity is that of being repeatedly under-taken in thought. In this case, the adverb ‘always’ provides the generativeaction function with its repetitive character, supplying it with the concep-tual dimension required in the generalizing task. The relevance of generat-ive action functions can be acknowledged by noticing that, in our example,generality is objectified as thepotential actionthat can be reiterativelyaccomplished.

But the current discursive formulation still needs more elaboration. Thestudents are not completely satisfied with it. The formulation is incomplete


according to their idea of cultural communicative standards required inthe practice of mathematics (cf. line 84). There is still a gap between thecurrent product and the envisioned task discussed in line 64.

4.3. Structural descriptors of the figure

What is it that allows the students to go further? The insertion, in thedialogue, of another deictic term – the word ‘horizontal’ (line 81; ‘hori-zontally’ in line 87) – is crucial in the search of new paths. Equipped withthe horizontal/vertical dichotomy (a dichotomy in which Lotman (1990)found one of the general traits in the way that human beings semiotic-ally mark their environment, as can be observed in cultures around theworld) the students’ generalizing task acquires a sharp and more preciseexpression. In fact, the words horizontal/vertical function as astructuraldescriptor of the figure. In particular, the word ‘horizontal’ allows thestudents to go a step further. They identify the number of the figure withthe amount of circles in its bottom row and start tackling the ‘positioning’problem, i.e. the problem consisting in relating the rank of the term in asequence to the numerical value of the term. (We shall come back to the‘positioning problem’ in the next section.) This is clear in line 124:

124 Anik: You (talking to Josh in paused segments although the talk seems more likea discursive strategy to organize her own thoughts with her words) have the samenumber of chips . . . (making a long horizontal gesture with her hand) . . . horizontally,like . . . the number of the figure . . . The number of the figure is 12 . . . soyou’re goingto have 12 on the bottom(looking attentively at him as if scrutinizing any sign ofacquiescence)

125 Judith: Yes (approving Anik’s statement)126 Anik: How do you . . . ? (still unsatisfied with what she said)127 Josh: The regularity (making a gesture with the hands)128 Judith: (Interrupting) The figure, if you know . . . you’ll have 12 on the bottom . . .129 Josh: If you know the regularity of the figure, it’ll be 12 on the bottom. Then it’s

always minus 1 for the top (Pause of some 8 seconds in which Anik and Judith thinkabout Josh’s proposal).

130 Anik: Yes . . . Yes . . . OK. Yes. I know. Yes, that’s it. But like how do you say that?131 Josh: Yes.132 Anik: OK (taking the page to read what they have done up until now)133 Josh: I don’t know.

Line 129 contains the verbalized form of what will be the skeleton of thewritten message later. This line, as many others, exhibits the verbal/mentaloperations that the students perform on the metaphorical Figure 12. Onemay say that they ‘touch’ the figure with words and so words now becomea new organ of sensation. Since the students are experiencing the generalthrough the particular, the geometric quality of the pattern remains their

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focus of attention. Their thinking moves around the rows of the figure,through linguistic terms playing deictic and generative action roles. As aresult, it is possible to see a fundamental contrast when it comes to expressgenerality through the algebraic formulan+ (n− 1) or 2n− 1 that cannotcapture, in the same manner, the sensual-spatial nature of the figures of thepattern.

All in all, a substantial gain has been achieved but, again, no satisfactoryelaboration is reached yet (cf. lines 126, 130, 131 and 133). The words‘top’ and ‘bottom’ (which are structural descriptors too, even though herethey convey a less precise descriptive meaning than the vertical/horizontalcouplet) do not sound appropriate for them. A more precise expression isoffered by Judith in line 142:

142 Judith: If it’s the figure it’ll always have the number . . . like if we say it’s Figure 12,you’ll have 12 on the bottom and then you’ll have one less on top vertically.

143 Josh: OK. Because . . .

4.4. The positioning problem

Generalization, as we have seen in the students’ dialogue, is undergoing aprocess of consecutive changes and refinements. Whilewhat to say seemsmore or less well agreed upon,how to say it is still under construction.To continue, they have to solve, in deeper detail, thepositioning problemwith which they were dealing before (line 124). By this we mean that upuntil now, the students succeeded in objectifying the numerical relationshipbetween the number of circles in the horizontal and vertical componentsin specific figures of the pattern and in the metaphorical Figure 12. Theynow have to relate it to a non-specific figure, saying ingeneral termswhichposition the figure occupies in the sequence.

157 Anik: (Talking to the teacher who was checking up on the work of a neighboringgroup) Miss, how do you say, like, the number of the figure? Like the . . . you know,like . . .

158 Judith: (Talking to the teacher) Since the same number of chips is the same numberof the figure, you know? It’s like in Figure 1 there’s 1, so the Figure 2 has 2 on thebottom, then if it’s the Figure 3, you’ll have 3 . . .

159 Anik: Then, like, we want to say like the number of the figure.160 Teacher: This(shows horizontal with her hand)is your number on the?161 Anik: (interrupting) Like the number of the figure is 4 here, right? (Shows Figure 4

on the sheet of paper made up of bingo chips).162 Teacher: OK.163 Anik: Well, there we want to say, we want to say that it’s like, the number of the

figure.


164 Teacher: OK. Well, the row on the bottom is the same as the rank of the figure(emphasizing the word rank).

165 Josh: So, the horizontal row . . . /[. . . ]180 Josh: The number of chips horizontally is the same . . .181 Judith: the same . . .182 Josh: the same . . . ?183 Judith: it’s the same as the rank of the figure. Rank (and then, trying to spell it, she

says) I think it’s rank, r-a-n-k /[. . . ]199 Anik: Then, after that, you have to say that you have less . . . you have one less

vertically.

Lines 157, 158 and 164 show three different rhetorical forms embodyingwhat Miller identifies as different genres of speech (Miller, 1984). Eachone provides individual views of the problem at hand emphasizing layeredaspects of it. In line 157, the problem is sketched in terms whose vagueformulation invites Judith to re-voice and precise it further for the teacher –which is what Judith actually does in the following line. Anik’s impreciseutterance is reflecting the difficulty the students have in expressing andobjectifying generality and requires the engagement of the teacher to fulfilwhat is not materially said in the message and to complete it with thestudent’s intended meaning. Strictly speaking, line 157 says, in fact, verylittle. One may say that the co-text (that which contains what is collateraland implicit) here says more than the text itself. Line 158 is, hence, anattempt to complete the message of line 157 and this is done in a more vividand dynamic form. The question in line 157 (although invisible for theteacher) holds the historical traits of the students’ previous joint experiencein dealing with the problem. In line 164, the teacher offers the answer tothe problem from the point of view of the fluent speaker. Then, with line164, comes the process of individual appropriation and(re)creation(Hall,1995) of the technical mathematical expressions. As we will see in thenext subsection, in the(re)creationof the teacher’s technical words, andin order to end up with the required message, the students will interweavethe teacher’s words with their own subjectivity and end up with a modifiedversion of the meaning of rank.

4.5. The final message

The final message was the following:

If I ask you to give me the amount of circles in Figure 12, there will be the same numberof circle (sic) horizontally that would be the same as the rank of the figure. And to have thevertical rank, you have to subtract 1 from the number of horizontal circles.

Some remarks need to be made concerning this message.First, thestructure: the message is divided into two parts that reflect

the geometrical configuration of the figures in the pattern. The first part

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explains how to find the number of circles in the horizontal line. Thesecond part tells the addressee (who was incorporated in the text by theexpression ‘If I askyou’) how to calculate the amount of circles in thevertical line. In this sense, the message keeps theperpendicularityof thecomponents of the described object. In other words, the semiotic identifiedstructure of the figures of the pattern induces the semiotic structure of theirgeneral description. There is still another way to say this. As mentionedin endnote 2, Peirce distinguished three kinds of signs: the icon, the indexand the symbol. The icon is a sign that resembles the represented object.‘An index is a sign which refers to the Object that it denotes by virtueof being really affected by that object.’ (Peirce, 1955, p. 102). Or, as Eco(1988, p. 75) says: ‘L’Index est un signe qui entretient un lien physiqueavec l’objet qu’il indique; c’est le cas lorsqu’un doigt est pointé sur unobjet’. The symbol refers to its object in an arbitrary manner, or as Peircesaid, ‘by virtue of a law’ (op. cit. p. 102). The analysis of the structureof the message written by the students and its structuralresemblancetoFigure 12 in the pattern makes it clear that the message, taken as a sign,belongs to Pierce’s icon category.

Second, the message relies on a particular example – that of Figure 12 –but, as previously stated, in a metaphorical way. It does not mean, however,that Figure 12 is merely an ornament. The allusion to Figure 12 in themessage is a token of the students’ struggle to talk about and objectifythe general term of the pattern. Figure 12 (which is no longer mentionedin the rest of the message and which could be omitted or replaced by theexpression ‘any figure’ without affecting the comprehension of the text)is that which makes it possible to talk about the still unspeakable generalterm of the pattern in the students’ evolving technical language. Figure 12is what ensures the students’ link between the particular and the general14.

Third, the word ‘rank’, first mentioned by the teacher is appropriatedand used by the students in a way that was not intended by the teacher. Thestudentsventriloquatethe teacher’s word. Ventriloquation is, according toBakhtin, the integration of somebody else’s words in our own discourse.This allows us totalk and tothink through the words of others. It is not amere copy of them, rather we adapt the words in terms of our pragmaticalneeds and particular expressive intentions (see e.g. Bakhtin, 1981, pp. 299,304, etc; Wertsch, 1991, p. 59; Todorov, 1984, p. 73). In theirventrilo-quation of the word ‘rank’, the students make it ahybrid construction(Bakhtin, 1981) supplying the teacher’s word with new meanings. Thus,the students talk about the rank of the vertical line. The confusion seemsto arise when the teacher contextually equated the rank of a figure with


the number of circles in its horizontal line. The rank then became sort ofsynonymous with ‘the size of the line’, that is, the number of circles in it.

It is important to notice here that our interest is not in finding whatwas ‘right’ and what was ‘wrong’ in the classroom. As Seeger (1991, p.153) wrote, ‘[m]athematical precision may not be the best model availableto steer the process of sense-making in the math classroom, especially inrelation to instructional discourse.’ Rather, we are trying to understand thediscursive meaning-making movements that lead to certain shared under-standings in the classroom. After all, although improperly used in termsof a fluent algebra speaking person, the word proved to be useful for thenovice students, allowing them to refine and carve with more precisionthe elements to experience and express the generalizing task. It is clearthat the word rank is also used in other human non-mathematical practices(e.g. in physical education competitions or music, where one talks aboutrankings). Its use in the study of patterns, however, has a different andparticular goal. The ‘transposition’ from one practice to another requiresan almost entire conceptual reconstruction. And this functions like the in-sertion of a new word in a language that one is starting to learn. First, youhear somebody using it and you want to do the same. You find that yournew word sounds strange and when you begin using it the pronunciationsurprises you more than your fluent speaking audience. Slowly, you dareputting it in a short phrase, then in another and again in another, some-times in a faulty form, but you are still understood so you keep going.The teacher’s utterance in line 164 (where the word ‘rank’ is first used) isechoed by Judith’s utterance. In fact, right after, we find her saying: ‘Then,it’s the same as the rank . . . the rank . . . it’s the same . . . it’s the same . . . ’.

Let us now turn to the expression of generality through standard algeb-raic symbolism.

5. EXPRESSING GENERALITY IN ALGEBRAIC SYMBOLISM

In the last part of the activity about generalization (question d), the studentshad to provide a mathematical formula for the general term of the pattern.It is impossible to go through all the dialogue (it contains 94 entries), solet us highlight the salient segments, focusing, as in the previous section,on the use of signs and the appropriation of their meanings in the emergingstudents’ algebraic thinking.

The students started discussing question d) in remarking that this isa question similar to the last question concerning the pattern about rect-angles discussed that morning by the teacher – as we saw previously, a

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question which led to the formulan× (n+ 2) and that was still written onthe blackboard:

271 Judith: (after reading the question) That’s what is on the blackboard.272 Josh: Yeah.273 Judith: That would be 12 and 11.274 Anik: I didn’t understand what you said.(Takes the paper to read the problem.)275 Josh: (Explaining to Anik) It’s the same thing as what we did on the blackboard.276 Anik: Wait a minute . . .(Takes the page from Judith and reads the problem)277 Josh: (Continuing his explanation after looking at the blackboard) but it’s n minus

1 . . .278 Anik: OK, yeah (and gives back the page to Judith).279 Josh: That’s n minus 1 . . .280 Anik: (repeating Josh’s utterance) n minus 1 . . . umm . . .281 Josh: (adding to Anik’s utterance) . . . times n. OK. Then it’s . . .282 Judith: (interrupting abruptly) No!283 Anik: (interrupting also) It’s not times. There is no times.284 Josh: Uuh . . . (he then tries to find something else and adds, using a reconciliatory

voice) n plus . . .285 Anik: That would be . . . 1 . . .286 Josh: No . . . n minus 1 . . . (inaudible)287 Anik: Wait! Let’s ask the teacher.(Raises her hand to call the teacher to come over

to the group.)

Our interpretation of the episode is that the understanding of ‘n’ was notthe same for all of the students. Josh and Judith seem to be satisfied withthe formula ‘n−1’, but Anik does not. While waiting the teacher’s arrival,the students kept discussinghow to write the formula:

299 Josh: Yes. You can just put n minus 1. No?300 Anik: Well, to have the . . . to have the . . . those that are . . . um . . . horizontal.

Anik is aware of the fact that the formula ‘n − 1’ does not include thecircles in the horizontal line, but she does not know how to include themin the symbolic expression. The teacher joins the group in line 310, whenthe students were still discussing in the spirit of lines 299–300:

310 Anik: OK. Miss? OK. On the other question,(takes the sheet)this here says . . . thishere (showing the page to the teacher) OK. We know what that means. But we don’tknow to say . . . like . . . figure . . . ummm . . .

311 Teacher: OK. Here the figure number n, all that that means there, is that it does notmatter what figure. It could be Figure 1, 2, 3, 4, 5

312 Anik: (interrupting and uttering with discouragement) We know that313 Teacher:. . . (continuing her explanation) 58 . . . this does not matter . . . it is not

important.314 Anik: OK (saying it without conviction).


When the teacher realizes that this explanation is not enough for the stu-dents, she continues:

315 Teacher: But you will always apply the same formula. On the board there, I wrote. . . let’s say, one formula . . . then it would work for all the figures. So, here, I wantyou to try to come up with a formula that you could use and that would work for allthe figures by replacing a letter. Okay? Then, let’s say even to go up to [Figure]150if you want. Because often you will require a formula. . . I won’t always be givingyou bingo chips, you know. OK? So, with the formula, you will be able to find outhow many bingo chips there are in Figure 25 without always having to write themall out.

The teacher is hence shifting from ‘n’ as a dynamic general descriptor ofthe figures in the pattern to ‘n’ as a generic number in a mathematicalformula. This requires a double-view of ‘n’ supported by different mean-ings: ‘n’ as an ordinal number, and ‘n’ as a cardinal number capable ofbeing arithmetically operated. This is discursively sustained by referringto the formula on the blackboard and supplemented now with pragmaticarguments (e.g. ‘often you will require a formula’). However, the centralquestion in line 310 was not stated clearly so the teacher, unaware of thegroup discussion and its specific difficulty, started an explanation aboutwhat she imagined the difficulty to be. Again, we do not mean that all thiswas wrong. To formulate a clear question is to have almost answered it tosome extent and, most of the time, this requires using the involved conceptswith great skill.

The teacher’s explanation in lines 311 and 315 then led Josh to propose‘n’ as the formula:

331 Josh: That would be n.332 Anik: (ignoring Josh) So, you could say: To know the . . . um . . . how many chips

to have vertically . . . you would subtract 1 from how many chips . . .(at this point,the teacher, seeing that Anik is getting close, makes a gesture to encourage her tocontinue)there are horizontally.

333 Teacher: OK. But now you have to say that without using words! Use letters! OK?334 Josh: You have to do 1 n minus . . .335 Teacher: OK. You’re getting it.(She departs from the group.)

A fundamental difference with the discussion held in question c) – a ques-tion to which we devoted the previous section – is that now no specificfigure is included in the dialogue. Indeed, Figure 12 has now been com-pletely evacuated from the discussion – apart from the ephemeral mentionmade by Judith in line 273 which was ignored by her group-mates. As wesaw, the metaphorical Figure 12 was a key element in the students’ objec-tification and construction of a conceptual relation between the particular

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and the general and a central source of sense-making in talking about thegeneral in natural language. Now the deictic terms (horizontal, bottom,etc.) seem to take over the role of Figure 12 and to become the basis forthe semiotization of the general in standard algebraic symbolism.

347 Anik: OK. (After a moment of silent reflection, she takes a deep breath and says:)We’ll just say that n is like . . . n is the chips that there are on the bottom (makinga horizontal gesture with the hand) . . . like . . . this is the number of chips that thereare on the bottom . . . OK.

348 Josh: Yes . . . well . . . no . . . well . . . OK.349 Anik: So (talking with difficulty, and making gestures with the forefinger she clearly

‘writes’ in the air what she says) . . . n minus one equals n.350 Josh: Yes.351 Anik: That’s it (satisfied).352 Judith: (who is in charge of writing the formula on the page, says) n minus one

equals n?353 Josh: ‘Cause n is the figure.354 Anik: Well, do you understand what it is that we said there, though?355 Judith: Yes.356 Anik: n minus 1. No (seeing that Judith is using brackets) . . . you don’t have to put

it in brackets.357 Judith: Well . . . no . . . (referring to the algebraic expression which she realizes is not

completely right.)358 Anik: OK.359 Judith: If n is the thing on the bottom . . . well . . . let’s say that it’s 4, well 4 minus

1 would be 3 . . . that wouldn’t get us back to n again. (The student notices that theequality does not work.)

360 Anik: Well . . . we’ll put another letter . . . n minus 1 equals c. Makes no differencewhat letter. You just have to put a letter. You don’t need to put it in brackets eitherbecause you’re not putting another equation beside it.

361 Judith: OK.362 Anik: You just have to write it next to it. You just have to (inaudible). OK? Write

(making a gesture with her hand as if she is writing) n minus 1 equals a letter. OK?

The students’ final formula ‘(n − 1) = a’ was crossed out and thenrewritten as ‘n− 1= a’.

Even though it was not easy, the students were able to produce a mes-sage in natural written language about the total number of circles in anyfigure. They could not, in contrast, produce the algebraic expression forthetotal number of circles in figure ‘n’. As long as Anik kept herself fromentering the territory of standard algebraic symbolism, she had the sensethat something was missing from Josh and Judith’s ‘n−1’ proposal. Whenshe finally crossed over to this foreign territory, she, like her group-mates,seemed to lose sight of the figure. And to fulfil the uncompleted symbolicformula, she transformed it into the algebraic expression:n−1= n. What,then, did ‘n’ mean for them?


In fact, the students provided ‘n’ with two different meanings. Josh andJudith took the formula that they had to construct as an object bearinga formal resemblance to the blackboard formula ofn × (n + 2). In thissense, they started from the formulan × (n + 2) and conceived the newone as a formula endowed with a mode of representation which is charac-teristic of theicons15. In assuming that the formula must look like the oneon the blackboard, they relied on an anticipated formal resemblance. Themeaning with which ‘n’ was provided was underpinned by a metonymicsemiotic process. That is, a process of substitution of terms consideredsimilar in some respect, in this casen + 2 becomesn − 1, and as Joshsuggested in line 281, the new formula would ben × (n − 1) instead ofn × (n + 2). This is for the first meaning of ‘n’. The second meaning isrelated to the students’ functional use of signs asindexes. This meaningwas endorsed by Anik.

To be able to explain this we have to refer back to lines 332, 334, 347and 349. What we notice is that it is in the junction of the sign system ofspeech and the bingo-chips-artefacts that ‘n’ found a niche out of whichto emerge. Indeed, in line 332 we found Anik saying: ‘To know the. . . um. . . how many chips to have vertically . . . you would subtract 1 from howmany chips . . . ’ , which later(line 334) acquired a more contracted formwhen Josh ventriloquated Anik’s utterance and said: ‘You have to do 1 nminus’. Then, Anik, in turn, counter-ventriloquated Josh’s utterance in line349. Using Peirce’s terminology, we can say that the letter ‘n’ is a sign of aspecific kind: it is anindexin the sense that it is pointing, like a gesture withthe forefinger, to the linguistic expressions contained in line 332. There is,indeed, a match between ‘you would subtract 1 from how many chips thereare . . . horizontally’ (line 332) and the left side of the students’ symbolicexpression ‘n−1= n’ (line 349, mediated furthermore by line 347: ‘We’lljust say that n is like . . . n is the chips that there are on the bottom’). Inother terms, the students’ production of symbolic expressions appear hereas contractions of words in speech16. Probably, because elementary schoolarithmetic is focused on finding answers (Kieran, 1989, p. 33), Anik wasled to ‘close’ the symbolic expression ‘n − 1’ by indicating itstotal. Thesign ‘n’, which pragmatically functions as an index, serves to point to thisresult. Hence, the final formula ‘n − 1 = n’. It is because of the indexicalnature of signs, as used by Anik, that the students found it completelylegitimate to replace ‘n’ by ‘another letter’, since, as Anik said, it ‘makesno difference’ (line 360).

To sum up, the students’ first meaning of signs was actually derivedfrom a metonymic process seeking to end up with a formula whose semi-otic characteristic was that of being an icon of the formula on the black-

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board. The second meaning was related to the students’ objectifying dis-cursive activity in which ‘n’ (functioning from a semiotic viewpoint asan index) replaced some key words in the students’ speech to generatealgebraic symbolic expressions. In the second case, the construction ofsymbolic expressions was heavily supported by the students’ discourse.Notice that Vygotsky found a similar phenomenon when children learnto write. Indeed, he remarked that the signs of the written language findsupport in the sign-words of speech and only later the signs of writtenlanguage acquire a certain autonomy:

Understanding written language is done through oral speech, but gradually this path isshortened, the intermediate link in the form of oral speech drops away and written languagebecomes a direct symbol just as understandable as oral speech. (Vygotsky, 1997, p. 142)

To conclude our remarks about the meaning of algebraic signs in Anik’sgroup, it may be stressed that although both meanings, the iconic andthe indexical, conflicted in lines 299 and 300, the students’ reached aconsensus regarding the final formula. This can be credited to the factthat Josh’s suggested formula (‘n − 1’) appears formally embedded in theformula ‘n − 1 = n’ (see e.g. Josh’s approving expression ‘Yes’ in line350). In subsequent classroom activities about generalization of patternsthe conflict in this small group of students between meaning of signs wasdeeper and the search of a common understanding took much longer.

6. THE NATURE OF STUDENTS’ EMERGENT ALGEBRAIC THINKING

What then can be said about the students’ algebraic thinking? Accordingto the semiotic analysis offered in the previous sections and in light ofour framework, algebraic thinking, we want to suggest, is the specificway in which the students conceptually acted in order to carry out theactions required by the generalizing task. That which makes ‘algebraic’the students’ thinking is the distinctiveness of the mathematical practicein which they engaged, namely, the investigation and expression of thegeneralterm of a pattern – something that may not be required at the levelof the arithmetic thinking. This is why, for us, the students were alreadythinking algebraically when they were dealing with the production of awritten message (Section 4), despite the fact that they were not using thestandard algebraic symbolism. But the particular way in which the studentsconceptually acted and that underpinned the emergence of students’ algeb-raic thinking was regulated by a socially established mathematical practicewhere the teacher plays a central role. This role was that of immersing andinitiating the students into the particularities of the signs and meanings


in which the practice of algebra is grounded. The students’ emergent al-gebraic thinking, as required in the generalization of patterns, appearedhence as the appropriation of a highly specialized kind of cognitive praxisrequiring a social use of signs and the understanding of their meanings toachieve specific expressions of generality.

Naturally, from the point of view of the individualquaindividual, everystudent brought forth his or her own unique contribution and achieved anunderstanding of signs and of algebraic techniques which, as evidencesuggests, were different in some aspects from one student to the other.Thus, in the quiz following the teaching unit about patterns, Anik wasable to comfortably express generalization in both natural and standardsymbolic algebraic language. Her written message did not have recourseto any metaphorical figure and, in contrast to the formula discussed inthis paper, her formula was now exact. This suggests that she was ableto produce and understand signs better than the first day of the activityand to meet the learning demands as prescribed by the curriculum17. Josh,in contrast, produced a written message based on a metaphorical figureand, instead of providing a symbolic formula, he carefully explained howto calculate the total number of circles through a metaphorical example.Judith, instead of writing a message for the total number of circles for anyfigure of the given pattern, gave a message that described the actions togo from one term of the pattern to the next and her formula was built onthe basis of signs as elementary indexes, much in the same form as thefirst-day activity discussed in this paper.

In accordance to our conception of algebraic thinking, we see the stu-dents’ mathematical behaviour aspersonalattempts to engage with theconcrete and the general in varied forms. Josh and Judith, who were able tofind the exact answer for the 10th and the 100th figure of the pattern in thequiz, seem to be more comfortable experiencing the general through thesemiotic strategy based on metaphorical figures and the use of signs thatthis requires. The signs with which the fabric of our intimate mathematicalexperiences is made up offer different semiotic possibilities and the un-avoidable aesthetic experience that accompanies our personal and uniqueencounter with the general may be lived in different forms. However, theforms taken by the personal attempts to engage with the concrete andthe general are not arbitrary. The strategy based on metaphorical figuresand those leading to the alphanumeric algebraic formula, as we saw, werejointly objectified by the students during the small-group work (where,e.g., processes of ventriloquation were of great importance). It does notmean, of course, that the mathematical enculturation achieved throughclassroom activities is a straitjacket for the mind. Rather we take classroom

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activities as a world of possibilities in which our intimate mathematicalexperiences occur. As Mikhailov rightly pointed out,In the individual’s mentality there is not a single phenomenon determined by social beingthat is not at the same time deeply personal. And, on the contrary, in the individual mental-ity each ‘only’ personal perception comes ‘only’ out of the social means of reflection, thechief of which is language. (Mikhailov, 1980, p. 198)

To summarize, instead of seeing algebra learning and the emergence ofalgebraic thinking as the direct imprint of the external environment onstudents’ minds, as empiricist or behaviorist pedagogical approaches withtheir corresponding models of direct transmission of knowledge contend,we see the emergence of algebraic thinking as resulting from the encounterbetween the individual’s subjectivity and the social means of semiotic ob-jectification. The varied forms taken by the students’ algebraic thinking inthe mathematical activity about patterns are seen as evidence of a complexprocess in which the students mesh personal and interpersonal tones withinthe limits of the contextual possibilities to actualize the mathematical prac-tices.

The dynamic process of knowledge acquisition happens to be led bythe irreducible tension between the individualquaindividual and the rangeof semiotic means of objectification offered by the historically constituteddomain of culture allowing experience to occur. As Bakhtin wrote in one ofhis earliest works found many years later in a dark, damp room in Saransk,Russia:An act of our activity, of our actual experiencing, is like a two-faced Janus. It looks intwo opposite directions: it looks at the objective unity of a domain of culture and at thenever-repeatable uniqueness of actually lived and experienced life. (Bakhtin, 1993, p. 2)

7. SYNTHESIS AND CONCLUDING REMARKS

Following a relatively recent research field in Mathematics Education whichdraws from cultural semiotics and focuses on the understanding of stu-dents’ use of signs and production of meanings, this article presented ananthropological approach to the didactic study of introductory algebra. Ourstudy of the students’ use of signs and production of meanings offers somenew insights, both on the practical and the theoretical level.

On the practical level, our investigation of the students’ use of signs(word-signs and letter-signs) sheds light on the processes of semiotic ob-jectification by evidencing the manner in which the students semioticizedand conceptualized the relation between the particular and the general. Itwas suggested that the objectification of generality was discursively elab-orated as apotential actarticulated on two key linguistic elements: the use


of deictic terms and adverbs of generative action out of which a concreteexample (here the Figure 12 of the sequence) functioned as a metaphor.The metaphoric Figure 12, crafted with terms like ‘rank’, ‘vertical’ and‘horizontal’, allowed the students to accomplish an apprehension and adiscursive representation of the general and provided them with a way forexpressing the general through the particular. In addition, the objectific-ation of the general in natural language proved to be fundamental to therise of the symbolic formula in that the symbolic formula appeared as con-tracted or abbreviated speech. An important result was to have identifiedthat the letter-signs resulting from this process of semiotic objectificationcorresponds to the category ofindexes.Nevertheless, algebraic formulas ascontracted speech have an intrinsic limitation: indeed, as we noticed in thestudents’ activity, algebraic formulas cannot seize the sensual aspect of thefigures and cannot be based on a paradigmatic example. In this sense, thepassage from a non-symbolic to a symbolic algebraic expression of gen-erality means two ruptures, one with the sensual geometry of the patternsand the other with the numerical feature of them. A deeper investigation ofthe very semiotic nature of indexes as used by novice students is an opendidactic problem that needs to be pursued in order to better understand thestudents’ first contact with symbolic algebra. A better comprehension ofthe semiotic nature of indexes should also enlighten the meaning of theequal sign that they tend to incorporate in the symbolic expressions.

On the theoretical level, in conceiving signs both as concrete compon-ents of mentation and vital parts of specific social semiotic means of objec-tification, our work provides some possibilities in which to elaborate newunderstandings about students’ algebraic thinking. We sketched here oneof those possibilities, based on the idea of thinking as a sign-mediated cog-nitive praxis. Within this context, the emergent students’ algebraic thinkingwas seen as the students’ insertion into an intellectual practice requiringa social use of signs and the understanding of their meanings to achievespecific expressions of generality. This kind of cognitive praxis was nothomogeneous. The students displayed different mathematical behavioursthat we took aspersonalattempts to engage with the concrete and thegeneral in varied forms and which resulted in differences of mastery toelaborate algebraic symbolic expressions.

In order to conclude this synthesis, let us hence come back to the two-fold aforementioned rupture underpinning the passage from a non-symbolicto a symbolic expression of generality, and to discuss it in light of theterritory of encounter of individual subjectivity and the social means of se-miotic objectification. This rupture encompasses a subjective experientialcomponent that, as we want to submit, includes an aesthetical aspect that

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we cannot neglect. The metaphorical figure and concrete examples takenfrom the pattern provide vivid figures of speech that are at odds with thefunctioning of the sober algebraic symbolism. Both correspond to radicallydifferent ways with which we can semioticize our world. To become afluent user of algebra (in the curricular sense) would then mean to acquirea set of tools and concepts allowing one to become conversant with thegeneral, the unknown and the variable in ways embedding aesthetical ex-periences which are different from those based on concrete or metaphoricallinguistic devices. For those students feeling comfortable with metaphor-ical figures of speech (like Figure 12), the rupture that we are mentioningmay somehow be similar to the one that a person in an art gallery maylive when, after experiencing paintings in the perspectival style of Pierodella Francesca and other painters of the Quattrocento, he or she movesto the next room exhibiting non-lifelike cubist paintings of Picasso andBraque. Although in both cases one may be ‘talking’ about e.g. portraits,the differences in the respective sign systems (e.g. lines, gradient colors,forms, etc.) and in their semiotic organization ensure a different order inthe ‘text’ and leads to different modes of expression and aesthetical exper-iences. Coming back to the didactic problem at hand, it may well be that infocusing on standard algebraic symbolism as the starting point to have thestudents thinking algebraically – whether in its alphanumeric form, in itsdisguised form as a ‘transitional language’ (see Section 1 above) or evenin a related disguised manner of using some hands-on manipulatives (see adiscussion of this in Radford and Grenier 1996) – we have narrowed downthe possibilities for the students to converse with the general. By excludingother forms of ‘texts’ based on different figures of speech, it would seemthat we have silenced some voices. The variety of algebras that we findthroughout the history of mathematics (see e.g. Radford, 2001) and whosetraces and sediments appear in one way or another in the phylogeneticallyconstituted practice of contemporary algebra in school, on the one hand,and the individual intellectual trends of our contemporary students, onthe other, suggest a variety of forms in which to semioticize the general.However, the celebration of cultural differences and the diversity of modesof thinking risk leading us to a sad picture of unrelated islands. In ourconception, algebraic languages (as natural languages do) might allow thestudents to interact between themselves and, in doing so, to elaborate newmathematical meanings and understandings.


ACKNOWLEDGEMENT

I wish to thank Paolo Boero for his comments on previous versions of thisarticle.

NOTES

1 Here, we intend ‘signs’ in a broad sense; they may be word-signs of written and oralnatural language, letters (e.g. ‘x’, ‘ n’) or even artefacts.

2 In Semiotics some authors make a distinction between signs and symbols. For in-stance, in Peircean semiotics, a symbol is understood as a specific kind of sign bear-ing (in contrast to the icon or the index) an arbitrary relation to its object (see Peirce,1955, p. 102; Parmentier, 1994, p. 6; Eco, 1988, p. 76). Vygotsky distinguishedbetween signs and marks, reserving in some cases the first term for marks havingbeen provided with meaning by the individual and often taking signs and symbolsindistinctly (see e.g. Vygotsky, 1997, p. 129, 139, 141passim). In this paper, we donot make any differentiation and take signs and symbols as synonymous.

3 For a detailed critique of didactic approaches to algebra centered on the study of thesyntactic aspect of the algebraic language see Nemirovsky (1994).

4 Vygotsky stressed many times this altering function of signs and tools in the psy-chology of individuals. In one passage, he wrote: ‘By being included in the processof behavior, the psychological tool alters the entire flow and structure of mentalfunctions. It does this by determining the structure of a new instrumental act just asa technical tool alters the process of a natural adaptation by determining the form oflabor operations.’ (Vygotsky, 1981, p. 137).

5 We take here activity in a neo-A.N. Leontievan sense (Leontiev, 1984), and weconsider the relation sign/activity after A.A. Leontiev’s elaboration (A.A. Leontiev,1981).

6 This conception of signs and other concrete artefacts leads to several practical im-plications. Since they are not conceived as sole accessories of the mind, educationalpractices (mostly at the primary level) no longer need to be oriented to the rapidabandonment of manipulatives and concrete artefacts to focus solely on mental orless concrete-based practices. This, naturally, does not mean that we will be confinedto walk around with a calculator in our pocket or with an abacus under our arm.What this means is that a different attitude towards signs and artefacts is a result ofthe theoretical position we are advocating.

7 Unfortunately, the isolation of the subject, the object and the act of symbolizingfrom the context encompassing the individuals’ semiotic activity often leads toreified analysis of language and discourse. In doing so, the individuals and theirrelations are sunk into oblivion and language and discourse become endowed witha kind of supernatural creative power. As Mikhailov noticed, ‘When formally ana-lysed, language hangs in the air, as it were, is deprived of its roots and becomesan independent object of research; the individual, whose tongue makes languagea living thing, is pushed into the background and forgotten.’ (Mikhailov, 1980, p.221).

8 A critique of this misleading and oversimplified interpretation of knowledge acquis-ition often ascribed to cultural approaches can be found in Waschescio, 1998.

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9 We argued elsewhere (Radford, 1998a), that the minimal and peripheral epistemo-logical role, that up to recent times has been given to speech, has a long-standingtradition in Western thought, where speech was considered (by Frege, 1971 amongothers) as something obstructing the space between the mental concept and its rep-resentation. Since ideas and concepts were conceived as a faculty of the solitairemind, speech was considered too noisy, ambiguous and illogical to reflect and ex-press the – supposedlyaphonic– nature of the concept and its silent paradise fromwhere voices and utterances were banished since the beginning. In the tradition ofWestern thought, the idea, the concept, the logos, are mute, voiceless, and can, atbest, be well dressed by signs if suitable dressings are found: this isgrosso modothe story of the quest of theUniversal Languagefrom the 17th century onwards.

10 The teacher is in charge of the instruction in the classroom.11 Although the reason for the quizzes is primarily related to our need of keeping track

of the mathematics classroom achievement (as typified by the curriculum) in orderto gain feedback for the design of the next teaching units, the quizzes bring forthcomplementary information to understand the problems related to meaning makingand sign use.

12 Those places where some lines are omitted will be indicated by ‘/[. . . ]’. Althoughthe duration of a pause in speech may obey different reasons, to give an idea of thecadence of the dialogue we will use ‘. . . ’ to indicate a pause of 3 seconds or more,and we will use ‘,’ or ‘.’ to indicate a pause of less than 3 seconds.

13 Metaphors have been recognized as important elements in the elaboration of newconcepts (see e.g. Sfard, 1994). A recent account is given in Presmeg (1998). Seealso Núñez 2000, Núñez et al., 1999.

14 There was a Greek mathematician by the name of Hypsikles who wrote a treatisecalledAnaphorikos(Manitius, 1888). This treatise – which had some influence onDiophantus’ work (see Radford, 1995) – has a few propositions about arithmet-ical progressions, and what is so particular about it is that Hypsikles proved themin a deductive manner:Anaphorikos, in fact, contains the first deductive knownproofs about arithmetical progressions and polygonal numbers. Hypsikles, in theabstract line-diagram upon which the deductive proof was based, inserted numericalexamples that may be considered redundant or superfluous. Yet, the recourse toconcrete numbers may also be taken, as in the case of our students, as an evidenceof Hypsikles’ efforts to talk about the general.

15 Algebraic expressions were among the usual examples that Peirce gave of icons(see Peirce, 1955, p. 104 ff).

16 Notice that the emergence of mathematical signs in the late Western Middle-Agesand the Early Renaissance followed a similar course. For example, the sign for thesquare-root was preceded by the word ‘root’ which was first writtenin extensoandlater abbreviated by a stylized form of its first letter ‘r’ (see Cajori, 1993). Notice,however, that we do not take this action in the students’ process of symbolizing asa recapitulative instance of phylogenetic development. We take it as a token of theimportance of indexes in semiotic strategies of objectification.

17 Although there is no room in this article to provide explanations about the growthof Anik’s understanding, let us notice that her improvement in sign use and theirmeanings may be related to her (direct and indirect) participation in the classroomgeneral discussions held after the groups completed their co-operative work.


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École des sciences de l’éducationUniversité LaurentienneSudbury, OntarioCanada P3E 2C6E-mail: [email protected]

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SIGNS AND MEANINGS IN STUDENTS’ EMERGENT ALGEBRAIC ... · The very semiotic nature of thinking and knowledge is addressed in Section 1, where we discuss some problems about signs